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I have two related questions about smooth complete algebraic curves (over $\mathbb{C}$).

  1. Does there exist a smooth complete algebraic curve $X$ that cannot be embedded as a complete intersection in $\mathbb{P}^n$ for any $n$ (nb: this is different from asking if every smooth curve in $\mathbb{P}^n$ is a complete intersection, which is of course false; e.g. the twisted cubic)? I expect that the answer is "yes", though it might be "no" for specific genera (and I'd be interested in known these genera). If you bounded $n$, then probably you could use the fact that the moduli space of curves is of general type for large genus to prove this.

  2. Fix a genus $g$. Does there exist some $n$ such that $\mathbb{P}^n$ contains a smooth genus $g$ curve as a complete intersection? I'm not really sure if the answer should be yes or no; if it is no, then I'd be interested in knowing which $g$ satisfy this.

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up vote 13 down vote accepted

1) The genus of a complete intersection of multidegree $(d_1,\ldots ,d_{n-1})$ in $\mathbb{P}^n$ is $g=1+\frac{1}{2} d_1\cdot \ldots \cdot d_{n-1}(\sum d_i-n-1)$ (just compute the degree of the canonical bundle). This gives very particular values for $g$: $1,\ldots ,5, 9, 10,12,16,\ldots $ . Any curve whose genus is not in this list cannot be realized as a complete intersection.

2) Even if $g$ is in that list, for $g>5$ a general curve of genus $g$ cannot be realized as a complete intersection, since the number of moduli of such complete intersection is smaller than $3g-3$ (the number of moduli of a general curve of genus $g$).

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Thanks! I feel a little foolish to have not realized (2)... – Robin Aug 29 '14 at 20:20
You write "$g=1,\ldots,5,9,10,12,16,\ldots$", but $g=2$ does not occur (while $g=0$ of course does). – Noam D. Elkies Aug 30 '14 at 4:26
Yes. Sorry I went too fast! – abx Aug 30 '14 at 5:59
Shouldn't 6 also be in that list, using a planar curve of degree 5? I also think 12 should not be in the list, whilst 13 should (using (3,2,2)) and so does 15 (as a curve of degree 7). – pbelmans Dec 27 '15 at 13:41
You are absolutely right. Sorry I didn't pay too much attention to the precise numbers -- I just wanted to indicate the principle. – abx Dec 27 '15 at 17:22

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